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arXiv · 2109.09333

Global existence and blow-up for the focusing inhomogeneous nonlinear Schr\"odinger equation with inverse-square potential

JinMyong An·JinMyong Kim·RoeSong Jang
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Abstract

In this paper, we study the Cauchy problem for the focusing inhomogeneous nonlinear Schr\"{o}dinger equation with inverse-square potential \[iu_{t} +\Delta u-c|x|^{-2}u+|x|^{-b} |u|^{\sigma } u=0,\; u(0)=u_{0} \in H_{c}^{1},\;(t,x)\in \mathbb R\times\mathbb R^{d},\] where $d\ge3$, $0 -c(d):=-\left(\frac{d-2}{2}\right)^{2}$. We first establish the criteria for global existence and blow-up of general (not necessarily radial or finite variance) solutions to the equation. Using these criteria, we study the global existence and blow-up of solutions to the equation with general data lying below, at, and above the ground state threshold. Our results extend the global existence and blow-up results of Campos-Guzm\'{a}n (Z. Angew. Math. Phys., 2021) and Dinh-Keraani (SIAM J. Math. Anal., 2021).

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JinMyong An, JinMyong Kim, RoeSong Jang. 2021-09-20. Global existence and blow-up for the focusing inhomogeneous nonlinear Schr\"odinger equation with inverse-square potential. https://arxiv.org/abs/2109.09333

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